Page 366 - Advanced Linear Algebra
P. 366
350 Advanced Linear Algebra
/
Of course, if ¢ / ¦ - is a bounded linear functional on , then
( ( ²%³
~ sup
))
%
%£ ))
The set of all bounded linear functionals on a Hilbert space / is called the
continuous dual space, or conjugate space, of / and denoted by / i . Note
that this differs from the algebraic dual of / , which is the set of all linear
functionals on / . In the finite-dimensional case, however, since all linear
(
)
functionals are bounded exercise , the two concepts agree. Unfortunately,
(
there is no universal agreement on the notation for the algebraic dual versus the
continuous dual. Since we will discuss only the continuous dual in this section,
no confusion should arise.³
The following theorem gives some simple reformulations of the definition of
norm.
be a bounded linear transformation.
Theorem 13.30 Let ¢/ ¦ /
1) )) ~ ) % sup )
))%~
2) )) ~ ) % sup )
))%
)
3 )) ~ s ) % ¸ inf ) ) ) for all % / ¹
%
The following theorem explains the importance of bounded linear
transformations.
be a linear transformation. The following are
Theorem 13.31 Let ¢/ ¦ /
equivalent:
)
1 is bounded
)
2 is continuous at any point % /
)
3 is continuous.
Proof. Suppose that is bounded. Then
) ) ) %c % ²% ) ~ ) c% ) )³ %c% ) ¦
)
)
)
as %¦% . Hence, is continuous at % . Thus, 1 implies 2 . If 2 holds, then
for any & / , we have
) ) ) %c & ~ ²% c& b% ) ³c ²% ³ ¦
as % ¦ & , since is continuous at % and % c& b% ¦ % as & ¦ % . Hence,
)
)
is continuous at any & / and 3 holds. Finally, suppose that 3 holds. Thus,
is continuous at and so there exists a such that
))% ¬ ) %
)
